Results 1  10
of
15
Small Covering Designs by BranchandCut
, 2000
"... A BranchandCut algorithm for nding covering designs is presented. Its originality resides in the use of isomorphism pruning of the enumeration tree. A proof that no 4 (10; 5; 1)covering design with less than 51 sets exists is obtained together with all non isomorphic 4 (10; 5; 1)covering desi ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
A BranchandCut algorithm for nding covering designs is presented. Its originality resides in the use of isomorphism pruning of the enumeration tree. A proof that no 4 (10; 5; 1)covering design with less than 51 sets exists is obtained together with all non isomorphic 4 (10; 5; 1)covering designs with 51 sets. 1 Introduction Let V be a set of elements of cardinality v and let k and t be integers such that v k t 0. Let K be the set of all ksubsets of V and T be the set of all tsubsets of V . For 1, a t (v; k; )covering design is a collection C of sets in K such that each t 2 T is contained in at least sets of C. A t (v; k; )covering design C is minimum if the cardinality of C is as small as possible. This cardinality is denoted by C (v; k; t). Covering designs have a long history and have applications in statistics, coding theory and combinatorics, among others. Numerous theorems give the value of a minimal covering design under certain assumptions on the p...
Improving bounds on the football pool problem via symmetry reduction and highthroughput computing
, 2007
"... The Football Pool Problem, which gets its name from a lotterytype game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. Using a combination of isomorphismpr ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The Football Pool Problem, which gets its name from a lotterytype game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. Using a combination of isomorphismpruning, subcode enumeration, and linearprogramming basedbounding, running on a highthroughput computational grid consisting of thousands of processors, we are able to report improved bounds on the size of the optimal code for this open problem in coding theory.
Improving Bounds on the Football Pool Problem by Integer Programming and HighThroughput Computing
"... The Football Pool Problem, which gets its name from a lotterytype game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. Using a combination of isomorphismpr ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The Football Pool Problem, which gets its name from a lotterytype game where participants predict the outcome of soccer matches, is to determine the smallest covering code of radius one of ternary words of length v. For v = 6, the optimal solution is not known. Using a combination of isomorphismpruning, subcode enumeration, and linearprogrammingbased bounding, running on a highthroughput computational grid consisting of thousands of processors, we are able to improve the lower bound on the size of the optimal code from 65 to
Trades and Defining Sets: Theoretical and Computational Results
, 1998
"... Given a particular combinatorial structure, there may be many distinct objects having this structure. When investigating these, two natural questions to ask are: . Given two objects, where and how do they differ? . How much of an individual object is necessary to uniquely identify it? These two q ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Given a particular combinatorial structure, there may be many distinct objects having this structure. When investigating these, two natural questions to ask are: . Given two objects, where and how do they differ? . How much of an individual object is necessary to uniquely identify it? These two questions are obviously related, with the first leading to the concept of a trade, and the second to that of a defining set. In this thesis we study trades and defining sets, in the context of t(v; k; ) designs. In our enquiries, we make use of both theoretical and computational techniques. We investigate the spectrum of trades, and prove an extant conjecture regarding this. Our results also suggest a more general version of this conjecture. A t(v; k; ) design where = 1 is called a Steiner design, and the related trades are called Steiner trades. In the case t = 2, we establish the spectrum of Steiner trades for each value of k, except for a finite number of values in each case. The conn...
Classification of designs with nontrivial automorphism groups
 J. Combin. Designs
"... Published online in Wiley InterScience (www.interscience.wiley.com). ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Published online in Wiley InterScience (www.interscience.wiley.com).
There are 270,474,142 Nonisomorphic 2(9, 4, 6) Designs
 J. COMBIN
"... We enumerate the 2(9; 4; 6) designs and find 270,474,142 nonisomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between 1 and 360. Out of these designs, 19,489,464 are simple and 2,148,676 are decomposable. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We enumerate the 2(9; 4; 6) designs and find 270,474,142 nonisomorphic such designs in a backtrack search. The sizes of their automorphism groups vary between 1 and 360. Out of these designs, 19,489,464 are simple and 2,148,676 are decomposable.
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.
A computer approach to the enumeration of block designs which are invariant with respect to a prescribed permutation group
, 1997
"... ..."
Enumeration of 2(12, 3, 2) Designs
"... A backtrack search with isomorph rejection is carried out to enumerate the 2(12; 3; 2) designs. There are 242 995 846 such designs, which have automorphism groups whose size range from 1 to 1536. There are 88 616 310 simple designs. The number of resolvable designs is 62 929; these have 74 700 noni ..."
Abstract
 Add to MetaCart
A backtrack search with isomorph rejection is carried out to enumerate the 2(12; 3; 2) designs. There are 242 995 846 such designs, which have automorphism groups whose size range from 1 to 1536. There are 88 616 310 simple designs. The number of resolvable designs is 62 929; these have 74 700 nonisomorphic resolutions.
Enumeration of 2(9,3,λ) Designs and Their Resolutions
"... We consider 2(9,3,λ) designs, which are known to exist for all λ ≥ 1, and enumerate such designs for λ = 5 and their resolutions for 3 5, the smallest open cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047 732, respectively. The designs are obta ..."
Abstract
 Add to MetaCart
We consider 2(9,3,λ) designs, which are known to exist for all λ ≥ 1, and enumerate such designs for λ = 5 and their resolutions for 3 5, the smallest open cases. The number of nonisomorphic such structures obtained is 5 862 121 434, 426, 149 041, and 203 047 732, respectively. The designs are obtained by an orderly algorithm, and the resolutions by two approaches: either by starting from the enumerated designs and applying a cliquefinding algorithm on two levels or by an orderly algorithm.